Abelian group algebras of finite order

Introduction. A group G of finite order n and a field F determine in well known fashion an algebra GF of order n over F called the group algebra of G over F. One fundamental problem(') is that of determining all groups H such that HF is isomorphic to GF. It is convenient to recast this problem somewhat: If groups G and H of order n are given, find all fields F such that GF is isomorphic to HF (notationally: GF'HF). We present a complete solution of this problem for the case in which G (and thus necessarily H) is abelian and F has characteristic infinity or a prime not dividing n. The result, briefly, is that F shall contain a certain subfield which is determined by the invariants of G and H and the characteristic of F. 1. Multiplicities. If G is abelian of order n and F is a field whose characteristic does not divide n, the group algebra GF has the structure